Muhammad was born in Mecca in 570 AD. Two hundred years earlier Emperor Constantine made Christianity the official religion of the Roman Empire. Mecca was at the edge of the Empire. Yes, there were Christian churches there. There were synagogues there too, as well as pagan temples (which is what the Kaaba in Mecca was before Muhammad conquered Mecca).
This is a common misconception. The Edict of Milan was granted by Emperor Constantine in the West and Licinius Augustus in the East in 313 granting religious freedom throughout the Roman Empire. In addition, the Edict of Milan ordered the restitution of property confiscated from Christians.
Christianity had to wait many years later under a different emperor to become the official religion of the empire.
As to Mecca, it is not quite fair to describe the pre-Mohammedan Kaaba as a pagan temple. It was a focus of Arab unity and had areas devoted to the worship of all the Arab tribe in the area, including both Jewish and Christian tribes. Yes, most of its floor and wall space were devoted to pagan cults, but that is not the whole story.
There is a tradition passed down among Arab Christians that when Mohammed destroyed all the pagan idols in the Kaaba, he found Torah scrolls, which he returned to their Jewish Arab owners, and he also found icons of Christ and the Theotokos, which he did not destroy, but allowed to remain in the Kaaba along the the meteorite fragment his followers associate with Abraham.
Please henceforth get your Imperial and Church history straight: St. Constantine (we Orthodox honor both him and his mother St. Helen with the title “Isapostolos” usually Englished as “Equal-to-the-Apostles”) did not make Christianity the official religion of the Empire. He joined his co-Emperor in legalizing Christianity and moved the capital to a city which was consciously built as a Christian city (Constantinople, or New Rome as he called it, had no pagan temples, but many Christian churches). From edict of Milan in 313 until the proclamation of (Nicene) Christianity as the state religion of the Empire in 380 by St. Theodosius, the Empire had no state religion, which Christianity, Judaism and Greco-Roman paganism all being religio licita (though the latter two came under increasing restrictions during that period — the reign of Julian the Apostate, when Christianity was again persecuted excepted).